The Line and the Vesica Piscis

A second point breaks the solitude of the first. Between the 2 lies an interval, and the shortest path across it is the straight line, the figure of the Dyad. Give each point its own circle, of the one radius that joins them, and the 2 circles cut each other in a lens: the vesica piscis, the first enclosed field in geometry, from which Euclid draws the triangle on the first page of the Elements.

I. The Figure

Τὸ Σχῆμα

Euclid's second definition is as spare as his first: γραμμὴ δὲ μῆκος ἀπλατές, "a line is breadthless length". The ends of a line are points, and a straight line is one that lies evenly with the points on itself.¹ The first 2 postulates grant the ruler: a straight line may be drawn from any point to any point, and a finite straight line may be extended continuously in a straight line.¹ Point, line, ruler and compass: with these the whole of plane geometry is built.

The line is the Dyad's figure because it needs 2 points and nothing else. Aristotle reports the doctrine of those who "reduce everything to numbers, and say that the formula of 'line' is the formula of 2", some of them making 2 the Ideal line itself.² Sextus Empiricus sets the same correspondence in the Pythagorean order of magnitudes: the point under the Monad, the line under the Dyad.³ Two points fix a direction and a distance. They don't yet enclose anything. The Temple's numerology says the same of the number: 2 is the interval before enclosure, the pair that has departed from unity while a fuller configuration awaits its third term.⁴

Within the circle of Layer 1 the line has a special office. A straight line through the centre, ending on the circumference in both directions, is the diameter, and Euclid's definition adds that it bisects the circle.¹ Proclus records that Thales was the first to state this: the diameter divides the circle into 2 equal parts.⁵ The Dyad cuts the Monad's figure in half. It's the first division, and the halves are equal.

II. The Construction

Ἡ Κατασκευή

Euclid's first proposition builds an equilateral triangle on a given line, and the way he does it is the way the vesica is born.⁶ The construction is the Dyad's own: 2 points, 2 circles, 1 radius.

1. Set 2 points, A and B. The distance between them is the radius of everything that follows.
2. With centre A and radius AB, describe a circle. It passes through B.
3. With centre B and radius BA, describe a second circle. It passes through A. The 2 circles are equal, and each runs through the centre of the other.
4. The circles cut at 2 points, C above the line and D below it. The lens enclosed between the arcs is the vesica piscis. Its long axis CD stands at right angles to AB and bisects it.

The measures fall out at once. The width of the lens is AB, the radius r. Its height CD is r√3, because ABC is an equilateral triangle of side r and its altitude is r√3⁄2, doubled. Each of the 2 bounding arcs is exactly one third of its circle, 120°.⁷ The ratio of height to width, √3 : 1, can't be written as a fraction of whole numbers. Archimedes bounded it from both sides with 265 : 153 below and 1351 : 780 above when he needed √3 to measure the circle.⁸

III. Number and Form

Ἀριθμὸς καὶ Σχῆμα

The line is the simplest figure that can be measured, and measuring it is where the Dyad shows its character. Two lengths can be compared: the double, 2 : 1, is the first ratio, and the Temple's numerology notes that number begins to speak through comparison exactly here.⁴ One line against another gives the octave, 2 : 1, the first consonance of the Tetractys in Layer 10. The straight line is also the figure of extension in a single dimension: length without breadth, as Euclid says, which is the Dyad's one direction out of the Monad's none.

The vesica is the Dyad enclosed. It's the field that 2 equal powers share when each reaches as far as the other's centre, and it's the only region that belongs to both circles. The next layer shows what the field contains: the 2 crossing points and the 2 centres are the corners of 2 equilateral triangles set base to base, and the line CD that joins the crossings is the first line the compass alone can set perpendicular to another. The Dyad's lens is where the Triad is waiting.

IV. In the Ancient Witnesses

Παρὰ τοῖς Ἀρχαίοις

The construction is as old as the Elements, about 300 BCE, and the figure within it is older than its Latin name. Vesica piscis, "the bladder of a fish", is the name medieval builders and later writers gave to the pointed oval; the geometry itself is Euclid I.1 with the triangle left out. In Gothic architecture the figure appears as the vertical aureole enclosing a seated figure in majesty, and as the frame of windows and portals; the rose window in the south transept of Lincoln Cathedral is a standard example.⁹ The pointed arch that carries those cathedrals is 2 arcs of the same construction set base to base.

The Pythagorean table of opposites, as Aristotle reports it, runs in pairs: limit and unlimited, odd and even, one and plurality, right and left, male and female, resting and moving, straight and curved, light and darkness, good and bad, square and oblong.¹⁰ Ten pairs, every one of them a Dyad, and "straight and curved" among them. The line and the circle are the figures of that pair, and the vesica is what happens when the curved is made to measure the straight.

V. The Zevist Reading

Ἡ Ζευϊστικὴ Ἀνάγνωσις

The Temple's teaching of the soul begins its own count at 2. The left is the receptive pole and the right the directing pole; Ida and Pingala, the black current and the red, are the 2 channels that flank the central path.¹¹ Two equal circles, each reaching to the other's centre, are the exact figure of that pair: neither current may swallow the other, and each must reach as far as the other's source. The field they share is the vesica, and in the Temple's teaching that field is where balance is won. Astarte's emblem expresses the balanced activity of Ida and Pingala and the awakening of Sushumna between them.¹² The axis CD, rising through the lens at right angles to the line of the 2 centres, is the figure of Sushumna before it's named: the central path drawn by the 2 that flank it.

The Dyad carries creation and destruction together in the Temple's core meanings, and the vesica shows why.⁴ Move the 2 centres apart and the lens thins and vanishes; move them together and the circles merge into 1 and the field is lost. Only at the exact distance of 1 radius does the form appear. Right relation is a measure, and Ma'at is the keeping of it. Light and Darkness, the first of the Temple's Antithetical Powers, operate in harmony when bound by Ma'at and in destruction when bound by Izfet; the 2 circles bound by 1 radius are harmony drawn with a compass.¹³

The straight line is the figure of command passing from one point to another: intention to action, the understood purpose to its outward enactment. The Temple's guide to the numbers in practice gives the Dyad this very pair.¹⁴ A line is drawn from A to B and not from anywhere to anywhere. Direction is the Dyad's gift to the Monad.

VI. Measure

Τὸ Μέτρον

VII. Sources and Reading

Πηγαί

1. Euclid, Elements I, Definitions 2 to 4 and 17, Postulates 1 to 2. Greek (Heiberg), Perseus · English (Heath, ed. Joyce).
2. Aristotle, Metaphysics Z 11, 1036b12 to 17. Greek · English (Tredennick).
3. Sextus Empiricus, Against the Physicists II (= Adversus Mathematicos X) 278. Greek, Scaife Viewer.
4. Temple of Zeus, 2 · The Dyad.
5. Proclus, Commentary on the First Book of Euclid's Elements, on Definition 17 (Friedlein 157.10 to 11), translated by Glenn R. Morrow (Princeton, 1970). Morrow translation.
6. Euclid, Elements I.1. Joyce, Proposition I.1.
7. Eric W. Weisstein, "Vesica Piscis", MathWorld. mathworld.wolfram.com.
8. Archimedes, Measurement of a Circle, Proposition 3, in T. L. Heath, The Works of Archimedes (Cambridge, 1897). archive.org.
9. James Stevens Curl, A Dictionary of Architecture and Landscape Architecture, "vesica piscis" (Oxford University Press). Oxford Reference.
10. Aristotle, Metaphysics A 5, 986a22 to 26. Greek · English (Tredennick).
11. Temple of Zeus, Red, White, and Black and The Pythagorean Star and the Two Orientations.
12. Temple of Zeus, Astarte's Symbol: Balance and the Living Center.
13. Temple of Zeus, Light and Darkness: The Cosmic Forces Under Ma'at and Izfet.
14. Temple of Zeus, How to Use the Numbers in Magick & Meditation.